In this video, we will learn how to
write and solve one-step multiplication and division equations in a variety of
questions, including word problems. Before starting these questions,
we’ll familiarise ourselves with the key vocabulary we will be using. Some of the key vocabulary or
definitions we will be using are as follows. Firstly, let’s consider what we
mean by a linear equation. These are equations that give a
straight line when plotted on a graph. For example, 𝑥 plus two is equal
to five. 𝑥 minus seven is equal to
three. Two 𝑥 is equal to 10. And 𝑥 divided by three is equal to
Note here that the equations
contain four different operations: addition, subtraction, multiplication, and
division. For the purpose of this video, we
will focus on those equations involving multiplication and division. This video will also only deal with
one-step equations and not two-step or multiple-step equations that you might see
later. A one-step equation is an equation
that requires only one calculation to solve it. In this particular video, this will
once again only involve multiplication and division.
During this video, we will discuss
reciprocal operations or inverse operations. These are operations that can undo
a calculation. The reciprocal or inverse of adding
is subtracting. The reciprocal of multiplying is
dividing, and vice versa. It is this key point that we will
focus on in this video. We know that four multiplied by
three is equal to 12. We also know that 12 divided by
three is equal to four.
Dividing both sides of the first
equation by three gives us the second equation. This is because the threes on the
left-hand side would cancel as three divided by three is equal to one. And we’re left with four is equal
to 12 divided by three. Remember, whatever you do to one
side of an equation, you must do to the other. We have to perform the same
operation on both sides of the equal sign as this ensures that the equation
continues to be balanced. We mentioned at the beginning of
the video that a key part will be solving equations. This involves finding the value of
𝑥 or the unknown that makes the equation true.
We will now look at some examples,
the first of which involve solving a one-step linear multiplication equation.
Anthony bought five bars of
chocolate of the same kind, each of the same price. If 𝑓 is the cost of each chocolate
bar, solve five 𝑓 equals 25 to determine 𝑓.
We’re told in the question that
Anthony buys five bars of chocolate with the same value. The letter 𝑓 represents the cost
of one bar. Therefore, the cost of five bars is
equal to five 𝑓. Our equation is equal to 25. Therefore, the total cost of the
five bars equals 25. In this question, we’re not given
the units. So we don’t need to worry if the 25
is in cents, pence, or any other currency.
We have been asked to solve the
equation to determine 𝑓. This is an example of a one-step
equation as we only need to carry out one calculation to solve the equation. Remember, when solving an equation,
you need to perform the same operation to both sides. This keeps the equation
balanced. In this case, we’ll divide both
sides of the equation by five. We do this because multiplying by
five and dividing by five are reciprocal or inverse operations.
Five 𝑓 divided by five is equal to
𝑓. The fives cancel as five divided by
five is equal to one. 25 divided by five is equal to
five. This means that the value of 𝑓 in
the equation five 𝑓 equals 25 is five. In the context of this question,
the cost of each chocolate bar is five units.
We will now look at a second
example where we need to write and then solve a one-step equation.
It takes James five times longer
than it takes Olivia to get to work. If James’s journey is 70 minutes,
write an equation for the time 𝑥 it takes Olivia to get to work. Then solve the equation.
In this question, we need to write
an equation and then solve it. We’re told in the question that it
takes Olivia 𝑥-minutes to get to work. It takes James five times
longer. Five multiplied by 𝑥 is equal to
five 𝑥. Therefore, it takes James five 𝑥
minutes to get to work. We’re also told that James’s
journey time is actually 70 minutes. This means that we have a linear
equation five 𝑥 is equal to 70.
As we have now written the
equation, we can move on to the second part of the question, which is to solve
it. We know that James’s journey time
is 70 minutes. And we need to solve the equation
five 𝑥 equals 70 to work out Olivia’s journey time. This is a one-step equation as we
only need to carry out one calculation to solve it. Remember, we need to perform the
same calculation or operation to both sides of the equation. In this case, we will divide by
five. Dividing by five is the inverse or
reciprocal of multiplying by five.
Five 𝑥 divided by five is equal to
𝑥 as the fives cancel. 70 divided by five is equal to
14. If we couldn’t work out this
calculation mentally, we could use the short division bus stop method. Seven divided by five is equal to
one remainder two. And 20 divided by five is equal to
four. Therefore, 70 divided by five
equals 14. The solution of the equation five
𝑥 equals 70 is 𝑥 equals 14. We can therefore conclude that it
takes Olivia 14 minutes to get to work.
We can check this answer by
considering James’s time and multiplying five by 14. As 70 divided by five was 14, we
know that five multiplied by 14 must be equal to 70. Multiplication and division are the
opposite or reciprocal of one another. James’s journey time was 70
minutes, which is five times longer than Olivia’s of 14 minutes.
We will now look at a third example
where we will set up an equation involving two variables.
Holly paints one garden chair in 12
minutes. Write an equation for the number of
chairs 𝑐 that she could paint in ℎ-hours.
We are told in this question that
Holly can paint one chair in 12 minutes. However, we’re interested in the
number of chairs that she can paint in ℎ-hours. Let’s firstly consider how many
chairs Holly could paint in one hour. We know that 60 minutes is equal to
one hour. We need to work out the number of
chairs that can be painted in 60 minutes.
We could begin to count up in
twelves. Holly would be able to paint two
chairs in 24 minutes as 12 plus 12 is 24. Adding another 12 gives us 36
minutes. So she can paint three chairs in
this time. Holly could paint four chairs in 48
minutes and five chairs in 60 minutes. You might however have noticed
immediately that 12 multiplied by five is equal to 60. Either way, we can see that Holly
can paint five chairs in 60 minutes or one hour.
We now need to write an equation
for the number of chairs 𝑐 that she can paint in ℎ-hours. If Holly can paint five chairs in
one hour, she could paint 10 chairs in two hours, 15 chairs in three hours, and so
on. In ℎ-hours, she would be able to
paint five ℎ or five multiplied by ℎ chairs. Our equation is therefore 𝑐 is
equal to five ℎ. We could then substitute in values
for ℎ and 𝑐. We could substitute in values for ℎ
to work out the number of chairs painted in a certain number of hours or substitute
in a value for 𝑐 to work out the number of hours it would take to paint a certain
amount of chairs.
Our next example involves a
real-world context involving fractions.
A rectangle’s width is one-sixth of
its length. Given that the rectangle’s width is
nine inches, determine its length.
We will answer this question by
firstly drawing a diagram and then setting up a one-step linear equation. Let’s consider a rectangle with
width 𝑊-inches and length 𝐿-inches. We’re told in the question that the
width is one-sixth of the length. The word “of” in mathematics means
multiply. So 𝑊 is equal to one-sixth
multiplied by 𝐿. This in turn can be written as 𝑊
equals one-sixth 𝐿 or 𝐿 divided by six.
In this particular question, we’re
told that the rectangle’s width is nine inches. We can substitute this into our
equation so that nine is equal to one-sixth 𝐿. This is the same as nine is equal
to 𝐿 divided by six. In order to solve this equation, we
need to perform the same operation on both sides of the equal sign. In this case, we will multiply by
six as multiplying by six is the opposite or inverse of dividing by six.
On the left-hand side, six
multiplied by nine or nine multiplied by six is equal to 54. On the right-hand side, the sixes
cancel. And we’re just left with 𝐿. As 𝐿 is equal to 54, we can
conclude that the length of the rectangle is 54 inches. We can check this answer by working
out one-sixth of 54. As this is equal to nine, which was
the rectangle’s width, we know that our answer of 54 inches is correct.
We’ll now look at our final example
of writing and solving a linear equation.
I take a number and divide it by
three and then divide by six. The result is two. What is the number?
We can solve this problem in lots
of ways. We will look at setting up a linear
equation and also using function machines. We will begin by letting 𝑛 be the
number. Our first step is to divide this
number by three. We usually write this when dealing
with algebra as 𝑛 over three. We then need to divide this answer
by six. This is the same as multiplying by
one-sixth as dividing by number is the same as multiplying by the reciprocal of that
number. Multiplying the numerators and the
denominators gives us 𝑛 over 18 or 𝑛 divided by 18. Dividing a number by three and then
by six is the same as dividing the number by 18.
We’re told in the question that the
result or answer is two. Therefore, 𝑛 divided by 18 equals
two. We can solve this equation for 𝑛
by multiplying both sides of the equation by 18. This is because the reciprocal of
dividing by 18 is multiplying by 18. And we must perform the same
operation to both sides. On the left-hand side, the
eighteens cancel, leaving us with 𝑛. Two multiplied by 18 is equal to
36. The number that we started with is
therefore 36. We can check this answer by firstly
dividing 36 by three. This gives us 12. Dividing 12 by six gives us
two. This means that our answer of 36
As mentioned at the start, an
alternative method here would be to use function machines. We are beginning with an input of
𝑛, dividing by three, and then dividing by six and getting an output of two. The reciprocal or inverse of
dividing by six is multiplying by six. Dividing by three has an inverse of
multiplying by three. If we know that the output is two,
we can calculate the input by firstly multiplying by six and then multiplying by
three. Two multiplied by six is equal to
12. 12 multiplied by three is equal to
36. This confirms that our input number
𝑛 was equal to 36.
We will now finish this video by
summarising the key points. As well as the definitions and
vocabulary that we looked at at the start of this video, we need to remember the
following to solve one-step equations. Firstly, we need to identify the
reciprocal or inverse operation. For example, to solve four 𝑥
equals eight, we have to divide by four to cancel multiplying by four. This is because multiplication and
division are reciprocal operations.
Secondly, we need to ensure that we
do the operation to both sides. This ensures that the equation
remains balanced. Finally, it is important to check
your answer by substitution. The solution to the equation four
𝑥 equals eight is 𝑥 equals two. We can check this by substituting
two back into the equation. Four multiplied by two is equal to
eight. Remembering these three key points
will help us solve any one-step equation involving multiplication and division.